BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Marc Lackenby (University of Oxford - UK) DTSTART:20221104T160000Z DTEND:20221104T170000Z DTSTAMP:20260423T022934Z UID:GEOTOP-A/27 DESCRIPTION:Title: Knot theory and machine learning\nby Marc Lackenby (University of Ox ford - UK) as part of GEOTOP-A seminar\n\n\nAbstract\nKnot theory is divid ed into several subfields. One of these is hyperbolic knot theory\, which is focused on the hyperbolic structure that exists on many knot complement s. Another branch of knot theory is concerned with invariants that have co nnections to 4-manifolds\, for example the knot signature and Heegaard Flo er homology. In my talk\, I will describe a new relationship between these two fields that was discovered with the aid of machine learning. Specific ally\, we show that the knot signature can be estimated surprisingly accur ately in terms of hyperbolic invariants. We introduce a new real-valued in variant called the natural slope of a hyperbolic knot in the 3-sphere\, wh ich is defined in terms of its cusp geometry. Our main result is that twic e the knot signature and the natural slope differ by at most a constant ti mes the hyperbolic volume divided by the cube of the injectivity radius. T his theorem has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume\, and the slope is corrected by terms corr esponding to short geodesics that have odd linking number with the knot. M y talk will outline the proofs of these results\, as well as describing th e role that machine learning played in their discovery.\n\nThis is joint w ork with Alex Davies\, Andras Juhasz\, and Nenad Tomasev.\n LOCATION:https://researchseminars.org/talk/GEOTOP-A/27/ END:VEVENT END:VCALENDAR